Generalized Barrier Functions: Integral Conditions & Recurrent Relaxations

Barrier functions constitute an effective tool for assessing and enforcing safety-critical constraints on dynamical systems. To this end, one is required to find a function h that satisfies a Lyapunov-like differential condition, thereby ensuring the invariance of its zero super-level set h≥ 0, This methodology, however, does not prescribe a general method for finding the function h that satisfies such differential conditions, which, in general, can be a daunting task. In this paper, we seek to overcome this limitation by developing a generalized barrier condition that makes the search for h easier. We do this in two steps. First, we develop integral barrier conditions that reveal equivalent asymptotic behavior to the differential ones, but without requiring differentiability of h. Subsequently, we further replace the stringent invariance requirement on h ≥ 0 with a more flexible concept known as recurrence. A set is (τ-)recurrent if every trajectory that starts in the set returns to it (within τ seconds) infinitely often. We show that, under mild conditions, a simple sign distance function can satisfy our relaxed condition and that the (τ-)recurrence of the super-level set h≥ 0 is sufficient to guarantee the system's safety.